$A$ wheel starting from rest is uniformly accelerated at $2 \, rad/s^2$ for $20 \, s$. It is allowed to rotate uniformly for the next $10 \, s$ and is finally brought to rest in the next $20 \, s$. The total angle rotated by the wheel (in radians) is ............

The graph between angular momentum $L$ and angular velocity $\omega$ is:

$A$ square lamina $OABC$ of side length $10 \ cm$ is pivoted at $O$. Forces act on the lamina as shown in the figure. If the lamina remains stationary,then the magnitude of $F$ is:

Two cylindrical hollow drums of radii $R$ and $2R$ and of a common height $h$ are rotating with angular velocities $\omega$ (anti-clockwise) and $\omega$ (clockwise) respectively. Their axes,which are fixed,are parallel and in a horizontal plane separated by $3R$. They are now brought into contact.
$(a)$ Show the frictional forces just after contact.
$(b)$ Identify forces and torques external to the system just after contact.
$(c)$ What would be the ratio of final angular velocities when friction ceases?

$A$ uniform rod of mass $m$ and length $\ell$ hinged at end $A$ is released from the horizontal position shown in the figure. Just after the rod is released:

Column $I$	Column $II$
$(A)$ Angular acceleration of $C$	$(P)$ $\frac{3g}{2}$
$(B)$ Angular acceleration of $B$	$(Q)$ $\frac{3g}{2\ell}$
$(C)$ Acceleration of $C$	$(R)$ $\frac{3g}{4}$
$(D)$ Acceleration of $B$	$(S)$ $\frac{3g}{\ell}$

State whether the following statements are True or False:
$(1)$ The angular acceleration of an object rotating with a constant angular velocity is always zero.
$(2)$ An object can have a moment of inertia without energy.
$(3)$ The radius of gyration of an object is a constant quantity.
$(4)$ $A$ figure skater spins faster when they pull their arms in because their moment of inertia decreases.